Seifert manifolds admitting partially hyperbolic diffeomorphisms

0.29 MB·

by  Andy Hammerlindl

SEIFERTMANIFOLDSADMITTINGPARTIALLYHYPERBOLIC
DIFFEOMORPHISMS
ANDYHAMMERLINDL,RAFAELPOTRIE,ANDMARIOSHANNON
7
1
0
2
ABSTRACT. Wecharacterizewhich3-dimensionalSeifertmanifoldsadmittran-
n
sitivepartiallyhyperbolicdiffeomorphisms.Inparticular,acirclebundleover
a
J ahigher-genussurfaceadmitsatransitivepartiallyhyperbolicdiffeomorphism
ifandonlyifitadmitsanAnosovflow.
5

]
S 1. INTRODUCTION
D
This paper deals with the problem of classification of partially hyperbolic
.
h diffeomorphisms in dimension 3. This program was initiated by the works of
t
a Bonatti-Wilkinson([BoW])andBrin-Burago-Ivanov([BBI]). Theseresultswere
m
also motivated by an informal conjecture, due to Pujals, which suggested that
[ therewerenonewtransitivepartiallyhyperbolicexamplestodiscoverindimen-

1 sion3(see[BoW,CHHU,HP3]).
v Thefirsttopologicalobstructionstotheexistenceofpartiallyhyperbolicdif-
6
feomorphismscomesfrom[BI]wheretheexistenceofpartiallyhyperbolicdif-
9
1 feomorphisms in S3 is excluded. Other topological obstructions related with
1
theexistenceofpartiallyhyperbolicdiffeomorphismsin3-manifoldswithfun-
0
. damentalgroupwithpolynomialgrowthcanbederivedfromtheworkof[BI];
1
see also [Pa]. In [HP , HP ], the first two authors provided a classification of
0 1 2
7 partially hyperbolic diffeomorphisms in3-manifolds with solvable fundamen-
1
talgroupgivinganaffirmativeanswertoPujals’conjectureinsuchmanifolds.
:
v Notice that these include Seifert fiber spaces whose fundamental group is of
i
X polynomialgrowth,inparticular,circlebundlesoverthesphereandthetorus.
r EventhoughPujals’conjectureturnedouttobefalse([BoPP,BoGP,BoGHP]),
a
thereisstill(atleast)onepertinentquestionwhichremainsopenandisinthe
spiritofthisconjecture.
Question1. Ifa3-manifoldwhosefundamentalgroupisofexponentialgrowth
admitsapartiallyhyperbolicdiffeomorphism,doesitadmitanAnosovflow?
Since the examples in [BoPP, BoGP, BoGHP] are obtained starting with an
Anosov flow, they do not provide a negative answer to the previous question.
Date:January6,2017.
R.P.andM.S.werepartiallysupportedbyCSICgroup618. R.P.wasalsopartiallysupportedby
MathAmSud-Physeco.
1
2 A.HAMMERLINDL,R.POTRIE,ANDM.SHANNON
Thisquestionseemshardinviewthatforthemomentwedonotknowwhich3-
manifoldsadmitAnosovflowsandmoreorlessthesameobstructionsweknow
fortheexistenceofAnosovflowsareobstructionsfortheexistenceofpartially
hyperbolicdynamics.
This paper grew out of the motivation to answer the question in a specific
familyofmanifolds: circlebundlesoverhighergenussurfaces,ormoregener-
ally, Seifert fiber spaces. The simplest sub-family where the question of exis-
tenceoftransitivepartiallyhyperbolicdiffeomorphismswasunknownwasΣ ×
g
S1whereΣ isaclosedsurfaceofgenusg ≥2.ByaclassicalresultofGhys([Gh])
g
weknowthatsuchamanifolddoesnotadmitAnosovflows.
We deal in this paper with partially hyperbolic diffeomorphisms in Seifert
fiber spaces. For notational simplicity, and for the convenience of the reader
notfamiliarwithSeifertfiberingsinallgenerality,westartbystudyingthe(ori-
entable)circlebundlecasewhichismoreelementary,thoughmostofthemain
ideasextendratherdirectly totheSeifertfibersetting(othersdo requiresome
furtherstudyandwefinishthepaperwithasectionwhichextendstheresultsto
themoregeneralsetting). Thestrategyistoshowthatthedynamicalfoliations
thatapartiallyhyperbolicdiffeomorphismcarriesare(afterisotopy)transverse
tothecirclefibers. Thisallowsustoprovidesomeobstructionsonthetopology
ofthecirclebundlethatweexposelater.Letusmentionthatthisworkalsomo-
tivates the study of horizontal foliations on Seifert fiber spaces to understand
partially hyperbolic dynamics and this is very much related with the study of
representationsofsurfacegroupsinHomeo(S1), aratheractivesubjectwhose
interesthasbeenrenewedintherecentyears(seee.g. [Bow,Mann]andrefer-
encestherein).
Theconsequenceofourmainresultswhichiseasiesttostateis:
Theorem. IfacirclebundleoverasurfaceΣofgenus≥2admitsatransitivepar-
tially hyperbolic diffeomorphism, then it admits an Anosov flow. In particular,
Σ×S1doesnotadmittransitivepartiallyhyperbolicdiffeomorphisms.
TheobstructioncanbestatedintermsoftheEulernumberofthefiberbun-
dleandsaysthatacirclebundleoverahighergenussurfaceadmitsatransitive
partiallyhyperbolicdiffeomorphismifandonlyifitsEulernumberdividesthe
EulercharacteristicofΣ.SeeTheoremAbelow.
IntheAnosovflowcase,thecorrespondingstatementforΣ×S1followsfrom
the classification result of Ghys [Gh] (see also [Ba] for the extension to Seifert
fiber spaces). The proof we give here does not provide a classification, but is
differentevenintheAnosovflowcase. Theresultsinthispaperwerepartially
announcedin[HP ].
3
1.1. PreciseStatementsofResults.
1.1.1. Circlebundles. ConsiderManorientablecirclebundleoveranorientable
surfaceΣofgenusg ≥2.Seesubsection2.4forprecisedefinitions.
SEIFERTMANIFOLDSADMITTINGPARTIALLYHYPERBOLICDIFFEOMORPHISMS 3
Every circle bundle over Σ can be obtained from Σ×S1 by removing a solid
torusoftheformD×S1 (withD ⊂Σatwodimensionaldisk)andregluingbya
mappreservingtheverticalfibersandgivingrisetoanew3-manifoldM which
isacirclebundleoverΣinatrivialway. Thenumberofturnsameridianofthe
form ∂D×{t} gives around a fiber after being sent by the gluing map is called
theEulernumberofthebundleandisdenotedbyeu(M)(see[CC,Chapter4of
BookII]or[Hat]formoredetailedinformationaboutthisconcept).
TheunittangentbundleT1ΣoverΣiswellknowntosatisfyeu(T1Σ)=χ(Σ)
whereχ(Σ)denotestheEulercharacteristic.Also,itisdirecttoshowthateu(Σ×
S1)=0.
Themainresultinthiscontextisthefollowing:
TheoremA. LetM beanorientablecirclebundleoverΣ,anorientablesurfaceof
genus g ≥2, admitting a transitive partially hyperbolic diffeomorphism. Then,
thereexistsk∈(cid:90)suchthateu(M)= χ(Σ) ∈(cid:90).
k
Remark. ThisconditionontheEulernumberisequivalenttosaythatthecircle
bundleisafinitecoveroftheunittangentbundle.
Seesubsection2.1forthedefinitionofpartiallyhyperbolicdiffeomorphism.
Inparticular,werequirethatthetangentspaceTM ofM splitsintothreenon-
trivial bundles, one uniformly contracting, one uniformly expanded and the
third one satisfies a domination condition with respect to the other two. This
issometimescalledintheliteraturepointwisestrongpartialhyperbolicity. We
saythatadiffeomorphism f istransitiveifthereisapointx∈M whose f-orbit
isdense.
Adirectconsequenceisthefollowing:
Corollary1.1. LetM beacirclebundleoverΣ,asurfaceofgenusg ≥2,thenM
admits a transitive partially hyperbolic diffeomorphism if and only if it admits
anAnosovflow.
Proof. ThetimeonemapofanAnosovflowispartiallyhyperbolicandiftheflow
istransitiveandnotasuspension,thenthetimeonemapisalsotransitive. The
mainresultof[Gh]impliesthatanAnosovflowinacirclebundlemustbetransi-
tive(andnotasuspension),thus,thereverseimplicationistrivial. Thefirstone
followsdirectlysincebyliftingthegeodesicflowinnegativecurvaturetofinite
covers one can construct Anosov flows in all bundles with the corresponding
(cid:3)
EulernumbersadmittedbyTheoremA.
Therearemanyconditionsweakerthantransitivitythatalsoallowustoob-
tain the same result. These appear in the statement of Theorem 3.1 below, so
beyondtransitivitywealsoobtain:
TheoremB. LetM beacirclebundleoverΣ,asurfaceofgenusg ≥2,admitting
apartiallyhyperbolicdiffeomorphismwhich
4 A.HAMMERLINDL,R.POTRIE,ANDM.SHANNON
• eitherisdynamicallycoherent,or,
• ishomotopictoidentity.
Then,eu(M)= χ(Σ) ∈(cid:90).
k
See subsection 2.1 for a definition of dynamical coherence: it is related to
the integrability properties of the bundles involved in the partially hyperbolic
splitting.
1.1.2. Seifert fiber spaces. We now consider the more general case of partially
hyperbolic systems defined on Seifert fiber spaces. Quotienting each fiber of
a Seifert fiber space M down to a pointdefines a projectionfrom M to an ob-
jectΣwhichistopologicallyasurface,butwhichismostnaturallyviewedasa
2-dimensional orbifold [Sco, Cho]. In this paper, every orbifold discussed will
be2-dimensional. Apartfromafewknown“bad”orbifolds,everyorbifoldmay
beequippedwithauniformgeometrywhichiseitherelliptic,parabolic,orhy-
perbolic. EventhoughΣdoesnothavethestructureofasmoothmanifold,its
unit tangent bundle T1Σ is a well-defined smooth 3-manifold and the natural
projectionT1Σ→ΣdefinesaSeifertfibering. Onemayalsodefinethegeodesic
flowonT1Σ. Asisthecasewithsurfaces, thisflowisAnosovifandonlyifthe
orbifoldishyperbolic.
Ifanorbifoldisbad,elliptic,orparabolic,thenaSeifertfiberspaceoverthis
orbifoldhasoneofthemodelgeometriescorrespondingtohavingvirtuallynilpo-
tentfundamentalgroup[Sco]. Partiallyhyperbolicsystemsinthesegeometries
arealreadywellunderstood[HP ].Hence,weonlyconsiderhyperbolicorbifolds
2
here.TheoremsAandBthenhavethefollowinggeneralization.
TheoremC. LetM beaSeifertfiberspaceoverahyperbolicorbifoldΣsuchthat
M admits a partially hyperbolic diffeomorphism which is either transitive, dy-
namicallycoherent,orhomotopictotheidentity. ThenM finitelycoverstheunit
tangentbundleofΣ.
BarbotshowedthatsuchmanifoldsareexactlytheSeifertfiberspaceswhich
supportAnosovflows[Ba].
Corollary1.2. ASeifertfiberspaceoverahyperbolicorbifoldadmitsatransitive
partiallyhyperbolicdiffeomorphismifandonlyifitadmitsanAnosovflow.
Someofthemoststudiedorbifoldsaretheso-called“turnovers.”Aturnoveris
aspherewithexactly3exceptionalpointsadded.Seifertfiberingsoverturnovers
were the last family for which the Milnor-Wood inequalities were generalized
[Nai],andBrittenhamshowedspecificallythatthesemanifoldsdonotsupport
foliationswithverticalleaves[Bri]. Inthissetting,wemaythereforestateare-
sultwhichdoesnotrelyontransitivityoranydynamicalassumptionotherthan
partialhyperbolicity.
Theorem1.3. ASeifertfiberspaceoveraturnoveradmitsapartiallyhyperbolic
diffeomorphismifandonlyifitadmitsanAnosovflow.
SEIFERTMANIFOLDSADMITTINGPARTIALLYHYPERBOLICDIFFEOMORPHISMS 5
Corollary1.4. ThereareinfinitelymanySeifertfiberspaceswhichsupporthori-
zontalfoliations,butdonotsupportpartiallyhyperbolicdiffeomorphisms.
1.2. Organizationofthepaper. Section2introducessomeknownfactsthatwe
willuseinthecourseoftheproofofourmainresults. Insection3westatethe
result which establishes, under some assumptions (including being transitive,
or dynamically coherent) that the dynamical (branching) foliations carried by
partiallyhyperbolicdiffeomorphismsmustbe(homotopically)transversetothe
fibersofthecirclebundleandworkoutsomepreliminariesfortheproof. This
resultisprovedinsections4and5.Then,insection6weusethisfactstogivethe
obstructionsintheEulernumberofthebundletoadmitsuchpartiallyhyper-
bolicdiffeomorphisms. Finally,insection7weextendtheresultstothegeneral
Seifertfibercase.
Acknowledgements:WethankChristianBonattiforseveralremarksandideas
thathelpedinthisprojectandinparticularforhishelpintheproofofLemma
4.11. WealsothankHarryBaik, JonathanBowden, JoaquínBrum, KatieMann
andJulianaXavierforseveralhelpfuldiscussions.
2. PRELIMINARIES
2.1. Generalfactsonpartiallyhyperbolicdynamics. LetMbea3-dimensional
manifoldand f :M→M aC1-diffeomorphism.Wesaythat f ispartiallyhyper-
bolicifthereexistsaDf-invariantcontinuoussplittingTM =Es⊕Ec⊕Eu into
1-dimensionalsubbundlesandN >0suchthatforeveryx∈M:
(cid:107)DxfN|Es(cid:107)<min{1,(cid:107)DxfN|Ec(cid:107)}≤max{1,(cid:107)DxfN|Ec(cid:107)}<(cid:107)DxfN|Eu(cid:107).
Itisabsolutelypartiallyhyperbolicifmoreover,thereexistsconstants0<σ<1<
µsuchthat:
(cid:107)DxfN|Es(cid:107)<σ<(cid:107)DxfN|Ec(cid:107)<µ<(cid:107)DxfN|Eu(cid:107).
ItispossibletochangetheRiemannianmetricsothatN =1andwewilldoso.
See[Gou].
OnecallsEsandEuthestrongstableandstrongunstablebundlesrespectively.
These integrate uniquely and give rise to foliations Fs and Fu called respec-
tivelythestrongstableandstrongunstablefoliations([HPS]). TheleafofFs or
Fu throughapointx willbedenotedbyWs(x)orWu(x). ThecenterbundleEc
maynotbeintegrableintoafoliation(thoughbeingone-dimensionalitalways
admitsintegralcurves).
WhenthebundlesEs⊕Ec andEc⊕Euintegrateinto f-invariantfoliationswe
saythat f isdynamicallycoherent,inthiscase,Ec integratesintoan f-invariant
foliationobtainedbyintersection.Wewillsometimesconsideralift f˜toM˜,the
universalcoverofM andweshalluseX˜ todenotetheliftofX whateverX is.For
Y containedinametricspaceZ wedefineBε(Y):={z∈Z : d(z,Y)<ε}.
6 A.HAMMERLINDL,R.POTRIE,ANDM.SHANNON
We refer the reader to [HP ] and references therein for a more detailed ac-
3
count. Inparticular, foraproofofthefollowingfactfrom[BI]thatwewilluse
repeatedly:
Proposition2.1. Foreveryε>0,thereexistsC >0suchthatifI isanarcofF˜u
orF˜s then
volume(Bε(I))≥Clength(I).
Theproofofthispropositionreliesheavilyontheconstructionofbranching
foliationsperformedin[BI]whichwereviewnext.
2.2. Branchingfoliations. Inthissubsectionweintroducetheconceptofbranch-
ingfoliationsandsummarisetheresultsfrom[BI]wewilluse.
A branching foliation F on M tangent to a distribution E is a collection of
immersedsurfacestangenttoE sothat:
• every L ∈F is an orientable, boundaryless and complete (with the in-
ducedriemannianmetric)surface,
• everyx∈M belongstoatleastoneL∈F,
• notwosurfacesofF topologicallycrosseachother,
• it is invariant under every diffeomorphism of M whose derivative pre-
servesE andatransverseorientation,
• ifx →x andL isaleafofF containingx thenL →L (uniformlyin
n n n n
compactsets)andx∈L∈F.
Wewilluse:
Theorem2.2(Burago-Ivanov). Let f :M →M be a partially hyperbolic diffeo-
morphismsuchthatEs,Ec,Eu areorientableandtheirorientationarepreserved
byDf. Then,thereexistbranchingfoliationsFcs andFcu tangentrespectively
toEs⊕Ec andEc⊕Eu.
Remark. The diffeomorphism is dynamically coherent if and only if Fcs and
Fcuhavenobranching:eachpointbelongstoauniquesurfaceofthecollection
(see[HP ]).
3
Toapplyfoliationtheory,wewillneedthefollowingresultwhichisalsoone
ofthekeypointsbehindProposition2.1.AsymmetricstatementholdsforFcu.
Theorem2.3(Burago-Ivanov). Foreveryε>0thereexistsafoliationWεtangent
toadistributionε-closetoEs⊕Ec andacontinuousmaphε:M →M suchthat
d(hε(x),x)<εforallx∈M andsuchthatwhenrestrictedtoaleafofWεthemap
hεisC1ontoaleafofFcs.
Indeedonehas(see[HP ]andreferencestherein):
3
Fact2.4. Thelifth˜εofhεtoM˜ whenrestrictedtoaleafofW˜εisadiffeomorphism
ontoitsimage(aleafofF˜cs). Moreover,thesediffeomorphismsvarycontinu-
ouslyasonechangestheleaf.
SEIFERTMANIFOLDSADMITTINGPARTIALLYHYPERBOLICDIFFEOMORPHISMS 7
Fact 2.5. ThefoliationWε isReebless1forallsmallε([CC]). Inparticular,when
lifted to the universal cover, the space of leaves Lε of the foliation W˜ε is a 1-
dimensional, possibly non-Hausdorff simply connected manifold. Using the
mapshεanditslifth˜εtotheuniversalcoveronededucesthatthespaceofleaves
ofF˜cs isalsoa1-dimensional,possiblynon-Hausdorffsimplyconnectedman-
ifold.
2.3. Torusleaves. Inthissectionwereviewaresultof[RHRHU]showingthatin
ourcontextthebranchingfoliationsFcsandFcucannothavetorusleaves.No-
ticethatthereexistsanexamplein(cid:84)3wheresuchleavesdoexist[RHRHU ].The
2
followingisasimplifiedversionofamoregeneralresultfrom[RHRHU]whichis
enoughforourpurposes.
Theorem2.6(RodriguezHertz-RodriguezHertz-Ures). Assumethatapartially
hyperbolicdiffeomorphism f :M →M hasatorusT tangenttoEs⊕Ec,then,M
fibersoverS1withtorusfibers.
NoticethatthisimpliesthatifM doesnotfiberoverS1withtorusfibersthen
neither the branching foliation Fcs from Theorem 2.2 nor the approximating
foliationsWεgivenbyTheorem2.3canhavetorusleaves.
2.4. Circlebundles. Thepaperwillfirstdealwiththecirclebundlecaseforwhich
therearesomesimplificationsinthenotationsandforwhichsomeresultsare
easiertostate. Laterinsection7weshallremovethisunnecessaryhypothesis
andworkingeneralSeifertfiberspaces.
RecallthatacirclebundleisamanifoldMadmittingasmoothmapp:M→Σ
(in our case, Σ is a higher genus surface) such that the fiber p−1({x}) for every
x ∈ Σ is a circle and there is a local trivialization: for every x ∈ Σ there is an
open setU such that p−1(U) is diffeomorphic toU×S1 via a diffeomorphism
preservingthefibers.
Notice that the fundamental group of the fibers fits into an exact sequence
whichisacentralextensionofπ (Σ)by(cid:90):
1
0→π (S1)∼=(cid:90)→π (M)→π (Σ)→0
1 1 1
Recalling that, if the genus of Σ is g ≥2, then the fundamental group π (Σ)
1
admitsapresentation:
(cid:68) (cid:89)g (cid:69)
π (Σ)= a ,b ,...,a ,b | [a ,b ]=id ,
1 1 1 g g i i
i=1
oneobtainsthatπ (M)admitsapresentation:
1
π (M)=(cid:68)a ,b ,...,a ,b ,c | (cid:89)g [a ,b ]=ceu(M), [a ,c]=id, [b ,c]=id(cid:69),
1 1 1 g g i i i i
i=1
1
Seealso[HP3,Section5]formoredetailsonhowthisfollowsfromNovikov’stheorem.
8 A.HAMMERLINDL,R.POTRIE,ANDM.SHANNON
whereeu(M)istheEulernumberofthecirclebundle.See[CC,BookII,Chapter
4]formoredetailsontheEulernumberandthefundamentalgroupof M. Let
usjustendbynoticingthatthecenter ofπ (M)isthe(cyclic)groupgenerated
1
bythefundamentalgroupofthefiber,calledcintheabovepresentation.Asthe
center is a group invariant and in this case is isomorphic to (cid:90) one knows that
everyautomorphismofπ (M)willsendc toeitherc orc−1.
1
Finally,letuspointoutthatcirclebundlesoversurfacesofgenus≥2donot
fiberoverS1withtorusfiberssothatTheorem2.6willimplythatapartiallyhy-
perbolicdiffeomorphismonsuchanM cannothaveatorustangenttoEs⊕Ec
orEc⊕Eu.
2.5. Tautfoliationsoncirclebundles. LetM beacirclebundleandletF bea
foliationonM whichhasnotorusleaves.Weremarkthatweusetheconvention
herethatafoliationisaC0,1+-foliationinthesensethatithasonlycontinuous
trivializationcharts,butleavesareC1 andtangenttoacontinuousdistribution
(see[CC]).
Foliationswithouttorusleavesin3-manifoldsareexamplesofwhatisknown
astautfoliations. WewillstatearesultofBrittenham[Bri]asitappearsin[Ca]
(whereadifferentproofinthecaseofcirclebundlescanbefound). Weremark
thatBrittenhamresultisanextensionofanoldresultofThurstonthatwasalso
improvedbyLevitt(see[Le]). Theproofof[Le]usesthehypothesisthatF isC2
butmostoftheproof(exceptthepartwhereitisshownthattherearenovertical
leaves) canbeapplied intheC0,1+ case asthey only useresultsofputtingtori
in general position which are now available forC0-foliations [So] (see also the
proofof[Gh,Proposition2.3]).
We need some definitions, we say that a leaf L of F is vertical if it contains
everyfiberitintersects. Equivalently,thereexistsaproperlyimmersedcurveγ
inΣsuchthatL=p−1(γ).
A leaf L of F is horizontal if it is transverse to the fibers (this includes the
possibilityofnotintersectingsomeofthem).
Theorem2.7(Brittenham-Thurston). LetF beafoliationwithouttorusleaves
inaSeifertfiberspaceM. Then,thereisanisotopyψ :M →M fromtheidentity
t
such that the foliation ψ (F) verifies that every leaf is either everywhere trans-
1
versetothefibers(horizontal)orsaturatedbyfibers(vertical).
Ofcourse,itispossibletoworkinverselyandapplytheisotopytopinorderto
havethatF hasthepropertythateveryleafisverticalorhorizontalwithrespect
top◦ψ−1.
1
Remark. AfterTheorem2.7itmakessense,foratautfoliationF,tocallaleaf
verticalifithasaloopfreelyhomotopictoafiber,andhorizontalifithasnosuch
loop.Wewilladoptthispointofviewinwhatfollows.
SEIFERTMANIFOLDSADMITTINGPARTIALLYHYPERBOLICDIFFEOMORPHISMS 9
3. VERTICALLEAVES
Let M be an orientable circle bundle over an orientable surface Σ of genus
g ≥2.Letp:M→Σdenotetheprojectionofthiscirclebundle.
We consider f : M → M to be a partially hyperbolic diffeomorphisms such
thatthebundlesEs,Ec,Eu areorientableanditsorientationispreservedbyDf.
LetFcs,Fcu befixedbranchingfoliationsassociatedto f.
WewillnowsaythataleafLofFcs orFcu isverticalifitsfundamentalgroup
containsanelementofthecenterofπ (M)(i.e.,itcontainsaloopfreelyhomo-
1
topicinM toafiberofthecirclebundle).SeetheremarkafterTheorem2.7.
Westatethefollowingresultwhichwillbeprovedinthenextsections.
Theorem3.1. For f andMasabove,underanyofthefollowingassumptions,the
branchingfoliationsFcs andFcu havenoverticalleaves:
(1) f ischainrecurrent2,i.e.,ifU isanon-emptyopensetsuchthat f(U)⊂U
thenU =M;
(2) f isdynamicallycoherent,i.e.,thebranchingfoliationshavenobranch-
ing;
(3) f ishomotopictotheidentity;
(4) theactionof f onπ (Σ)ispseudo-Anosov;
1
(5) f isabsolutelypartiallyhyperbolic.
Initem(4)wearetakingintoaccountthat f∗:π1(M)→π1(M)preservesthe
generatorcorrespondingtothefibers,andthereforeinducesanactiononπ (Σ)
1
viathefiberbundleprojectionp.Thegeneralcase,where f maynotbedynam-
icallycoherentandtheactiononπ (Σ)isreducibleisstillopen,thoughwethink
1
thattheproofsweprovideshedsomelightonhowtoattackit. Noticethatre-
centlynon-dynamicallycoherentexampleshaveappearedinSeifertmanifolds
([BoGHP])butthesehavehorizontalbranchingfoliations.
Remark. Toproveitem(2)wegiveageneralstatementaboutverticalcsandcu-
laminationswhichworksevenifthediffeomorphismisnotdynamicallycoher-
ent(seeProposition4.12). Similarly,toprovepoint(4)weshowaslightlymore
generalstatementthatmaybeusefulincertainsituations(seeProposition5.2).
UsingTheorems2.7and2.3wegetthefollowingresultwhichiswhatwewill
usetoobtainourmaintheorem:
Corollary3.2. Underanyoftheconditions(1)-(5)itfollowsthatforeverysmall
ε>0thereexistsaprojectionpεisotopictopsuchthatforWε,theapproximating
foliationtoFcs,onehasthateveryleafistransversetothefibersofpε.
The proofs are symmetric on cs and cu, so we concentrate on the cs-case.
Item (2) is the most involved and will be proved separately in section 4. The
2
Thisincludesasparticularcaseswhenf istransitiveorvolumepreserving.
10 A.HAMMERLINDL,R.POTRIE,ANDM.SHANNON
restoftheitemswillbeprovedinsection5andtheirproofsareindependentof
section4withtheexceptionofitem(5).
3.1. Generalities. DefineΛcs⊂M tobetheunionofallverticalcs-leaves.
Proposition3.3. ThesetΛcs iscompact, f-invariantandΛcs(cid:54)=M.
Proof. The action of f on the fundamental group must preserve the center of
π (M) and therefore, as leaves of Fcs are sent to leaves of Fcs, those which
1
containaloophomotopictoafiberareinvariant.
ToshowthatΛcs isclosed,weworkintheuniversalcover,andnoticethatif
asequencex →x andL areleavesthroughx whichareinvariantunderthe
n n n
deck transformation generated by c (a generator of the center of π (M)) then,
1
the same holds for a leaf L which is the limit (in the sense of uniform conver-
genceincompactsets). Therefore,LalsobelongstotheliftofΛcs andcontains
x.
ToshowthatΛcs(cid:54)=Mwewillmakeanadhocdirectargument3.Analternative
proofcanbeobtainedbyusingtherestoftheresultsinthissection.
Wefollowanargumentborrowedfrom[Ba ]. Weworkintheuniversalcover,
2
wherethepropertiesofthebranchingfoliationimplythatthespaceofleavesL
isa1-dimensional(possiblynon-Hausdorff)simplyconnectedmanifoldwhere
π (M)acts(seeFact2.5).
1
IfoneassumesbycontradictionthatΛcs =M, oneobtainsthateveryleafis
fixedbyageneratorc ofthecenterofπ (M). Astherearenotorusleavesandc
1
commuteswitheveryotherelementofπ (M),itfollowsthateveryelementnot
1
inthecenterofπ (M)actsfreelyonL.Thisimplies,byaTheoremofSacksteder
1
(see[Ba ,Theorem3.3])thatL isalineandtheactionisbytranslations.
2
Moreover,ifonechoosestwoelementsg ,g inπ (M)suchthatneitherg ,g
1 2 1 1 2
noritscommutatorbelongtothecenterofπ (M),oneobtainsacontradiction
1
sincethecommutatorshouldacttrivially,butitcannothavefixedpointsinL.
(cid:3)
Thereaderonlyinterestedinitem(1)ofTheorem3.1cansafelyskiptosub-
section5.1foradirectproof.
To continue we will introduce the concept of quasi-isometrically embedded
submanifold. Given b >0 we say that a submanifold L of a manifold M is b-
quasi-isometricifforeveryx,y∈Lonehasthat:
d (x,y)≤bd (x,y)+b,
L M
whered denotesthemetricinducedbyM onLandd isthemetricinM.
L M
3
Noticethatiff isdynamicallycoherent,thisfollowsdirectlybyapplyingTheorem2.7.Indeed,as
Λcswouldbeaverticalfoliationonewouldbeabletoprojecttheleavestogetaonedimensional
foliationinthesurface,contradictingthatithasgenus≥2.